Preheat the oven to 375°F (190°C). Grease a 9×13 inch casserole dish with butter.
Bring a large pot of salted water to a boil. Add the sliced potatoes and cook for 5 minutes until slightly tender but not fully cooked. Drain and set aside.
In a large skillet over medium heat, melt the butter. Add the chopped onion and cook until translucent, about 3-4 minutes.
Add the minced garlic and cook for 1 minute until fragrant.
Sprinkle the flour over the onions and garlic, stirring constantly for 1-2 minutes to form a roux.
Gradually whisk in the milk, stirring continuously until the sauce thickens, about 3-4 minutes.
Remove the skillet from heat and stir in the cream of mushroom soup, sour cream, salt, black pepper, thyme, and paprika until smooth.
Add the chicken pieces to the sauce and stir to coat evenly.
In the prepared casserole dish, layer half of the sliced potatoes evenly on the bottom.
Spread half of the broccoli florets over the potatoes.
Pour half of the chicken and sauce mixture over the broccoli layer.
Sprinkle 1 cup of cheddar cheese and 1/2 cup mozzarella cheese evenly over the chicken layer.
Repeat the layering with the remaining potatoes, broccoli, chicken mixture, and cheeses.
Sprinkle the grated Parmesan cheese evenly over the top layer.
Cover the casserole dish with aluminum foil and bake in the preheated oven for 35 minutes.
Remove the foil and bake for an additional 15 minutes, or until the cheese is golden brown and bubbly and the chicken is cooked through (internal temperature should reach 165°F or 74°C).
Let the casserole rest for 5 minutes before serving.